The Sharingan is a watered down version of Rinnegan+Juubi, the reason Uchiha are so dark and such in my opinion is because like Naruto when the elder son was born he got traits from the Juubi (Sharingan) this is why they steal each others eyes, kill close friends to advance, the Juubi was a demonic beast..and demands blood for power
The elder son clearly did not have Sharingan but it was not a full Rinnegan, it was like an incomplete Rinnegan, over time the more the generations thinned out the Rinnegan aspect fully disappeared and all that was left is the Sharingan
Now Byakugan is rumored to be related to Sharingan, maybe it is maybe its not, if it is my theory is that some of the Uchiha did not follow this path of hatred and darkness and broke away from the clan and formed there own, even tho the darkness of Juubi influence is still kind of there, making the Hyuuga clan have two branches, the main and the ..slaves pretty much but they didn't steal each others eyes and crap..leading to the Sharingan eventually being totally watered out
It is not said that you need Sharingan to obtain Rinnegan..
The way to obtain Rinnegan is having Uchiha and Senju DNA, not the eyes themselves, so this means if Naruto for example, got Uchiha DNA he could technically possess Rinnegan and have those powers, but not Sharingan powers, where as if Sasuke got Senju DNA, he would have both Sharingan and Rinnegan powers, like Madara
This is why the So6P is only known to use Rinnegan powers and not EMS powers but Madara is able to use both, because So6P never had Sharingan..the Sharingan comes from Juubi
Why Byuakugan was thrown into the dust idk..its just ended up not being important to the main story